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Tag Archives: mathematics

Imagine a history book that examines the philosophical foundations of mathematics, specifically the quest that culminated in the years leading up to the First World War to establish all mathematical reasoning on a firm logical basis. That book would have a lot of ground to cover. It would have to disentangle some complex mathematics to present to the non-specialist in a meaningful way, as well as shed light on the manic, driven, fascinating characters behind this story, people like Bertrand Russell, Kurt Gödel, David Hilbert, and Ludwig Wittgenstein. Finally, it would need to give at least some light to the background historic scaffolding upon which this drama played out: the turn of the century, the First World War, the rise of Nazism, and interwar Vienna. At tall order for any book, let alone a comic book.

So now imagine that book as a graphic novel—moreover, as a graphic novel that succeeds at all these tasks. That’s what you’ve got with Logicomix, a complex, stirring, well-executed, multi-layer work that brings to life one of the most compelling chapters of mathematical and philosophical history. In a general sense, the graphic novel (which is hefty, weighing in at over 300 pages not counting the reference material at the end) could be considered a stylized biography of Bertrand Russell (1872-1970), 3rd Earl Russell, the British logician and grandson of the Prime Minister, who began his career with an attempt to bring logical rigor to all mathematical reasoning.

Beyond Russell’s stylized biography (stylized because the historical interactions in the graphic novel are artfully fudged for better dramatic effect), the narrative of Logicomix plays out on three levels. Level one is the primary chronological narrative, but level two is the fact that this primary narrative is presented as a lecture delivered by the aging Russell in America near the end of his career. Hecklers in the audience want to know whether Russell, who was famous for his conscientious objections during the First World War, will join them in protesting America’s entry into the Second. Russell promises them their answer in the lecture, and these interactions, as Russell summarizes his career and offers insights on the role of logic in human affairs, bookend the first level narrative and interrupt it occasionally as audience members get rowdy or impatient.

This first narrative—the series of chronological flashbacks forming Russell’s lecture—is the main medium of the story telling in Logicomix. We see Russell as a young, troubled child in an authoritarian home finding the basis of truth and certainty in mathematics. As a student in Cambridge, Russell becomes obsessed with the logical foundations of mathematics, catalyzed by the 1900 challenge of David Hilbert and using the new logical formalism of Gottlob Frege to establish mathematics on completely rigorous, firm foundations. This is the work he spends the first decades of his career on, collaborating with Alfred North Whitehead to produce their Principia Mathematica, which—as Russell recounts wryly—took over 200 pages to prove that 1 plus 1 equals 2.

If this sounds like the stuff of esoteric mathematics, it is. But the success of Logicomix is making the story—which depends on the mathematics—both accessible and engaging. It provides enough of the technical details for the reader to get a conceptual notion of set theory, upon which Russell’s work rested, and the damning implications of Russell’s paradox, which undermined these very foundations. The narrative continues, always through Russell’s eyes though his own work leaves the center stage, to explore Gödel’s incompleteness theorem, Wittgenstein’s Tractatus Logico-Philosophicus, and the rise and fall of the Vienna Circle on the eve of the Second World War.

It’s not quite history (as the authors admit they’ve altered the timeline a bit to make Russell have meetings with characters that he likely never met), but it is a sweeping and effective story of people and their ideas. It’s not quite philosophy or mathematics either, but there’s enough of both to make Logicomix intellectually rich and rewarding—from the logical puzzles themselves (boiled down to their conceptual themes) to exploration of philosophical approaches to mathematics, contrasting Gödel’s Platonic to Poincare’s inductive to Wittgenstein’s linguistic approach to the true meaning of mathematics and its relation to the physical world or the human mind. It’s a story with meat on its bones, executed in bright, clean, understated art that brings the characters and the locales to life without overshadowing the concepts it explores.

As with history of thought done well, the book is as much about the people as the ideas with which they wrestled. One of the primary themes in is the question of the sort of mind or personality it takes to devote a life to wrestling with the basics of logic. We see this most with Russell and the background of madness he worked and fought against, as well as in the periphery characters of Cantor, Frege, Gödel, and Hilbert. The close relationship between madness and logic—as well as questions of the place of logic in life—are explored by Russell himself in the course of his lecture and by the authors and artists of the book as they make their appearance (and interact with the reader) throughout in the third “meta” level of the narrative.

It is this third level of narrative—and the balance it takes to run an additional narrative overtop of Russell’s lecture and his chronological flashbacks—that pushes Logicomix in some of its most interesting directions. This meta narrative represents the self-referential nature of the book itself (nicely complimenting the theme of paradox in logic arising through self-reference, as in the case of Russell’s set theory paradox and Gödel’s incompleteness theorem): the authors and artists are characters in their own book, working in modern Athens to write about Russell and the logical foundations of mathematics. We are invited into their studio to witness the discussions between them as they work. In this way, we simultaneously receive additional background to what happens before and after the events of the novel, the rational behind their specific approaches, and what we as readers are supposed to take from the story. As a bonus, we learn a lot about ancient Greek tragedy as well, which, tied elegantly to the discussion of logic and madness at the book’s conclusion, brings the work to its poignant conclusion.

Self-reference does not work well in logic and mathematical proof, but it does quite nicely in literature (The Neverending Story, Gene Wolfe’s Peace, and The Princess Bride immediately spring to mind). There are other parallels to draw between the axiomatic formalism of mathematics and the rules and consistency that govern storytelling, but that is a post for another time. Suffice it to say, Logicomix is incredibly rewarding and opens to door to a host of further readings in history, mathematics, philosophy, and logic, aided and abetted by the helpful reference section at the end. Not many books I read merit the creation of an entire new shelf of “to read” books on Goodreads, but this one did.

Everyone knows that Pythagoras was an early Greek mathematician, that he proved the Pythagorean theorem, and that he was one of the first to glimpse our modern conception of the world– that the universe can be described by numbers. Everyone “knows” this, but is there actually any historical basis to these claims? What do we really know about Pythagoras and what he did, and how much of what is taught about him in math classes is actually myth? Apparently quite a bit, according to Alberto Martinez.

The Cult of Pythagoras could have as easily been titled The Myths of Pythagoras. Martinez, a historian of science at the University of Texas, Austin, convincingly argues in the first two chapters of this work that the foundation on which we’ve built the myth of Pythagoras and his accomplishments is very thin indeed. Martinez does what generations of math historians and popularizers of science have failed to do: drill down to the source material and examine what ancient authorities actually have to say about the man. What he finds is that the earliest accounts are vague, contradictory, and emphasize Pythagoras’s mythical attributes– his teachings as a religious figure and his reported miracles– as much as they do his mathematics. What fascinates Martinez is the way that these accounts have been distorted and magnified over the centuries until we get the Pythagoras of modern conception today: the veritable father of mathematics.

Pythagoras actually takes up only fraction of this book. The subtitle, “Math and Myths,” gives a better indication of the bulk of the work. Besides Pythagoras, Martinez debunks other famous myths from the history of mathematics. Gauss finding the sum of all integers from 1 to 100 during a grade school exercise. Euler getting imaginary numbers wrong. Galois’ tragic tale. The golden ratio popping up everyone where in nature and art and architecture. If the book was simply a historian of science plumbing the depths of the historical source material and making modern promulgators of these stories look foolish, it would be worth the admission alone.

But Martinez has a deeper program here. There’s a fundamental myth about mathematics that he uses many of these other minor myths to explode. And that is the Platonic conception of mathematics as something somehow independent of the physical world itself, existing beyond our own mental constructions. This is the perception of mathematics existing eternal and unchanging, of mathematical discovery as not inventing new systems but instead discovering truths that were there all along. What Martinez sees instead, when he looks at the history of mathematics, is the story of things being formalized and formulated, not discovered. In particular, Martinez examines the nature of imaginary numbers, the problem of dividing by zero, and the rules regulating multiplication by negatives. These are not mathematical properties written in stone, Martinez argues, though they’re often taught that way. They are instead conventions that developed slowly over time.

Against a mathematical Platonism on the one hand and a radical constructivism on the other, Martinez ventures into philosophy and poses his own system of mathematical pluralism. Some fundamental tenants of mathematics are true independent of human though. 2 + 2 will always equal 4, for instance, whether or not there is anyone around to see or discover this fact. But other mathematical principles are constructed, like William Hamilton’s quaternions. The problem is, Martinez doesn’t provide us with any way of distinguishing which portions of mathematics fall into which category. Are the principles of Euclidean geometry independent of human thought? Would the Pythagorean theorem hold for all right triangles, regardless of whether there were humans around to mentally construct them? Or does the construction of self-consistent non-Euclidean geometries argue against this? There’s fertile ground for philosophical speculation there, which I would have liked to have seen Martinez follow up on.

At the end of the book, Martinez returns to Pythagoras. Why is it so easy to hang accomplishments on this man’s name without any secure historical basis? Beyond mathematics, Martinez explains, Pythagoras also gets attributions from religion, new age thought, philosophy, alchemy, astronomy, and more. Here Martinez ventures into sociology, explaining how accomplishments (whether actual or not) tend to accrue to people who are already “famous.” The very paucity of real data regarding Pythagoras, Martinez concludes, makes him a sort of vessel in which all these attributes can be poured, a well-known cipher from antiquity for our own values that we wish to project into the past.

In sum, The Cult of Pythagoras, though the prose is in places is uneven and the book itself wanders in the multiple points it makes, is a powerful argument for expelling myth from the teaching of mathematics. The history of mathematics itself, based not on unfounded stories but on the real historical events and accomplishments, is far more interesting and compelling than the unhelpful myths that are propagated regarding mathematicians and the practice of mathematics itself. Martinez’s scholarship is grounded on what the texts actually tell us, and I heartily recommend to anyone teaching mathematics. The chapters on Pythagoras alone make this worth any mathematician’s bookshelf.